Electric signal filters are circuits which pass signals having frequencies of interest while rejecting or attenuating undesired signals. The range of frequencies relatively unattenuated by the filter is called the passband; the range of significant attenuation is called the stopband.
In many applications, the amount of variation in attenuation, or ripple, in the passband must be limited to a small amount, such as 1 decibel (dB). For a low-frequency passing ("low pass") filter, the passband is typically defined to be all frequencies attenuated less than 3 dB; the 3 dB frequency is thus referred to as the cutoff frequency. With a low pass filter, as frequency increases above the cutoff frequency, the attenuation response continues to drop off through a transition region to the stopband. The stopband is typically defined as including those frequencies experiencing some minimum attenuation, such as 40 dB. A filter is said to have a sharp cutoff if the range of frequencies within its transition region is relatively small.
A specific attenuation response does not completely describe the transmission properties of a filter. Another parameter of importance is the phase shift of the output signal relative to the input signal. In fact, the phase shift response is often of primary importance in applications such as digital communications where the input signal must be amplitude-limited. This is because even if the input signal's major frequency components are entirely within the filter's passband, the output signal may be distorted if the propagation delay for different frequencies through the filter is not constant. Constant time delay corresponds to a phase shift increasing linearly with frequency; hence the terms "linear phase shift filter" and "constant group delay filter" are interchangeable. Thus, sharp cutoff, minimum ripple, and constant group delay are all desirable filter characteristics in digital signal applications.
Filters can also be described in terms of their time domain properties such as rise time, overshoot, and settling time. These are also important considerations when filtering digital signals. Unfortunately, these optimum time domain response requirements often conflict with optimum frequency domain requirements. For example, the need for limited rise time and overshoot often conflicts with constant group delay and sharp cutoff requirements.
Various types of low-pass filters have been developed which optimize one or more of the aforementioned parameters. However, these filters invariably optimize one particular parameter of the frequency or time domain response at the expense of other parameters. For example, a Butterworth filter provides minimum ripple in the passband for a given number of poles, but it has a large transition region, i.e., its cutoff is not very sharp, and it has relatively poor group delay characteristics.
The Chebyschev filter allows greater ripple to occur throughout the passband, which has the effect of greatly improving the sharpness of the cutoff. A Chebyschev filter is specified in terms of the number of poles and allowable passband ripple; its amplitude response in the frequency domain may be expressed as ##EQU1## where N is the number of poles, C.sub.N is the Chebyschev polynomial of the first kind of degree N, .epsilon. is a constant that sets the passband ripple amount, f is frequency and f.sub.c is the desired cutoff frequency. Chebyschev filters are optimum in the sense that they require the fewest number of poles for a prescribed maximum ripple and sharpest possible cutoff. However, with a Chebyschev filter, relatively constant group delay is realized only when sharp cutoff is avoided. Furthermore, even an ideal Chebyschev filter exhibits about 26% overshoot in the time domain; in practice this may be 40% or more.
An elliptic filter distributes ripple in both the passband and the stopband; indeed, it can be proven that an elliptic filter has the sharpest possible cutoff for a given number of poles N. The amplitude response of an elliptic filter is given by: ##EQU2## where U.sub.N is a Jacobian elliptic function. Here, .epsilon. specifies the ripple amount across the entire range of frequencies. Unfortunately, the elliptic filter is less than ideal for many digital signal applications since the stopband cutoff is very steep. This, in turn, means that the time domain response will exhibit large overshoot and long settling time.
Bessel filters are generally recognized as maximally phase-linear. The Bessel filter's time domain characteristics are excellent, and its time domain response exhibits very little overshoot; in fact, it exhibits undershoot. However, this undershoot phenomenon means that successive data bits of different polarities may interfere with one another; this, in turn, means that extra bandwidth must be allocated for a Bessel-filtered signal.
Another type of filter, called a transitional filter, is a compromise between the Chebyschev and Bessel filters. Certain inductor-capacitor implementations of this filter can achieve sharp frequency domain cutoff with about 15% overshoot. However, generally useful active filter implementations with non-ideal operational amplifiers typically introduce delays which are not taken into account in the theoretical analysis. As a consequence, real-world implementations of transitional filters tend to exhibit greater overshoot and less out-of-band attenuation than the theoretical analysis would otherwise indicate. These problems can be alleviated somewhat with high gain-bandwidth amplifiers, but these are expensive and prone to oscillation if not carefully designed.
What is needed for digital signal filtering applications is an inexpensive, linear-phase filter with sharp cutoff that exhibits low overshoot in the time domain.